50th Anniversary of JTAM
SOME EXACT SOLUTIONS FOR OLDROYD-B FLUID DUE TO TIME DEPENDENT PRESCRIBED SHEAR STRESS
Muhammad Jamil
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan and
NED University of Engineering and Technology, Department of Mathematics, Karachi, Pakistan e-mail: jqrza26@yahoo.com
Corina Fetecau
Technical University of IASI, Department of Theoretical Mechanics, Romania Mehwish Rana
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
The velocity field and the shear stress corresponding to motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders are es-tablished by means of the Hankel transforms. The flow of the fluid is produced due to the time dependent axial shear stress applied on the boundary of the inner cylinder. The exact solutions, presented under a series form, can easily be specialized to give similar solutions for the Maxwell, second grade and Newtonian fluids performing the same mo-tion. Finally, some characteristics of the motion as well as the influence of the material constants on the behavior of the fluid are underlined by graphical illustrations.
Key words: Oldroyd-B fluid, velocity field, time-dependent shear stress, Hankel transform
1. Introduction
During the past two decades, viscoelastic fluids are considered to play a more important and appropriate role in technological applications in comparison with Newtonian fluids. Large industrial materials fall into this category, such as solutions and melts of polymers, soap and cellulose solutions, biological fluids, various colloids and paints, certain oils and asphalts. Thus, due to
diversity of fluids in nature, several models have been suggested in the litera-ture. Amongst these, rate type fluids have gained much popularity (Bird and Armstrong, 1987; Dun and Fosdick, 1974; Dun and Rajagopal, 1995; Oldroyd, 1950; Rajagopal and Kaloni, 1989; Rajagopal and Srinivasa, 2000). One of the most popular model for the rate type fluids is known as the the Oldroyd-B fluid model. The flows of Oldroyd-B fluid have been studied in much details, more than most other non-Newtonian fluid models and in complicated flow geometries. The Oldroyd-B fluid can be found frequently in the field of blo-wing and extrusion molding. Unfortunately, the response of polymeric liquids is so complex that no model can adequately describe their response. As the Oldroyd-B fluid can describe stress-relaxation, creep and the normal stress differences but it cannot describe either shear thinning or shear thickening, a phenomenon that is exhibited by many polymer materials. However, this model can be viewed as one of the most successful model for describing the response of some polymeric liquids.
The exact analytical solution for the flow of an Oldroyd-B fluid was given by Waters and King (1970), Rajagopal and Bhatnagar (1995), Fetecau (2003, 2004), Fetecau and Fetecau (2003, 2005). Other interesting results were obta-ined by Georgiou (1996) for small one-dimensional perturbations and for the limiting case of zero Reynolds number, unsteady unidirectional transient flows of the Oldroyd-B fluid in an annular pipe, and unsteady transient rotational flows of the Oldroyd-B fluid in an annular pipe are given by Tong and Liu (2005) and Tong and Wang (2005). Some simple flows and exact solutions of the Oldroyd-B fluid are also examined by Hayat et al. (2001, 2004). Wood (2001) has considered the general case of helical flow of the Oldroyd-B fluid, due to combined action of rotating cylinders (with constant angular velocities) and a constant axial pressure gradient. Following Rahaman and Ramkissoon (1995), he assumes that the velocity profiles have Taylor series expansions and uses this assumption to derive a second initial condition. Fetecau et al. (2007) obtained the velocity fields and the associated tangential stresses correspon-ding to some helical flows of Oldroyd-B fluids in a series form in terms of the Bessel functions. However, it is important to point out that all the above mentioned works dealt with problems in which the velocity is given on the boundary. To the best of our knowledge, the first exact solutions for flows into cylindrical domains, when the shear stress is given on the boundary, are those obtained by Waters and King (1970) for Oldroyd-B fluids. Similar solutions, corresponding to a time-dependent shear stress on the boundary, have been recently obtained in Fetecau et al. (2009a,b, 2010), Jamil et al. (2011), Nazar
In this note, exact solutions corresponding to the axial flow of an Oldroyd-B fluid in an annular region between two infinite circular cylinders are establi-shed. In order to produce flow of the fluid, the boundary of the inner cylinder is subject to a time-dependent axial shear stress. The solutions that have been here obtained tend to the similar solutions for the Maxwell, second grade and Newtonian fluids by taking appropriate limits. Finally, the influence of physi-cal constants on the velocity profile and the shear stress is shown by graphiphysi-cal illustrations.
2. Basic flow equations
The Cauchy stress T in an incompressible Oldroyd-B fluid is given by (Fetecau
et al., 2009a,b, 2010; Jamil et al., 2011; Nazar et al., 2011; Siddique and Sajid, 2011)
T= −pI + S
S+ λ( ˙S − LS − SL⊤) = µ[A + λr( ˙A − LA − AL⊤)] (2.1)
where −pI denotes the indeterminate spherical stress due to the constraint of incompressibility, S is the extra-stress tensor, L is the velocity gradient,
A = L + LT is the first Rivlin Ericsen tensor, µ is the dynamic viscosity
of the fluid, λ and λr are relaxation and retardation times, the superscript (·)⊤
indicates the transpose operation, and the superposed dot indicates the material time derivative. The model characterized by constitutive equations (2.1) contains as special cases the upper-convected Maxwell model for λr→0 and the Newtonian fluid model for λr→0 and λ → 0. In some special flows,
as those to be considered here, the governing equations for an Oldroyd-B fluid resemble those for a fluid of the second grade. For the problem under consideration, we shall assume a velocity field and an extra-stress of the form V = V (r, t) = v(r, t)ez S= S(r, t) (2.2)
where ez is the unit vector in the z-direction of the system of cylindrical coordinates r, θ and z. For such flows, the constraint of incompressibility is automatically satisfied. If the fluid is at rest up to the moment t = 0, then
V(r, 0) = 0 S(r, 0) = 0 (2.3)
In the absence of body forces and a pressure gradient in the axial direction, the balance of linear momentum and constitutive equation (2.1) lead to the relevant equations 1 + λ∂ ∂t τ (r, t) = µ1 + λr ∂ ∂t ∂ ∂rv(r, t) ρ∂ ∂tv(r, t) = ∂ ∂r+ 1 r τ (r, t) (2.4)
where ρ is the constant density of the fluid and τ = Srz is the shear stress that is different from zero.
Eliminating τ between Eqs. (2.4), we obtain the governing equation
λ∂ 2 ∂t2v(r, t) + ∂ ∂tv(r, t) = ν + α∂ ∂t ∂2 ∂r2 + 1 r ∂ ∂r v(r, t) (2.5) where α = νλr and ν = µ/ρ is the kinematic viscosity of the fluid. Partial differential equation (2.5), with adequate initial and boundary conditions, can be solved in principle by several methods, their effectiveness strictly depending on the domain of definition. In our case, the integral transforms technique presents a systematic, efficient and powerful tool. The Hankel transform can be used to eliminate the spatial variable.
3. Flow through an infinite annular region
Suppose that an incompressible Oldroyd-B fluid at rest is situated in the annular region between two infinite coaxial circular cylinders of radii R1 and R2(> R1). At time t = 0+ the inner cylinder is pulled along its axis with a
time-dependent shear stress
τ (R1, t) = f [t − λ(1 − e−λt)] t > 0 (3.1)
where f is a constant, while the outer one is held fixed. Due to shear, the fluid between cylinders is gradually moved, its velocity being of form (2.2). The governing equations are given by Eqs. (2.4)1 and (2.5) and the appropriate
initial and boundary conditions are
v(r, 0) = ∂v(r, 0)
and for t > 0 1 + λ∂ ∂t τ (r, t) r=R1 = µ1 + λr ∂ ∂t ∂v(r, t) ∂r r=R1 = f t v(R2, t) = 0 (3.3)
In order to solve this problem, we shall use the Hankel transforms. 3.1. Calculation of the velocity field
In order to determine the velocity field, let us denote by (Tong and Wang, 2005; Debnath and Bhatta, 2007)
vnH(t) = R2
Z
R1
rv(r, t)B(r, rn) dr n = 1, 2, 3, . . . (3.4)
the finite Hankel transform of the function v(r, t), where
B(r, rn) = J0(rrn)Y1(R1rn) − J1(R1rn)Y0(rrn)
rn being the positive roots of the equation B(R2, r) = 0 and Jp(·), Yp(·) are the Bessel functions of the first and second kind of the order p. Using Eqs. (3.3), and the known relation
B(R1, rn) = J0(R1rn)Y1(R1rn) − J1(R1rn)Y0(R1rn) = − 2
πR1rn we can prove that
R2 Z R1 r ∂ 2 ∂r2 + 1 r ∂ ∂r v(r, t)B(r, rn) dr = 2 πrn ∂v(R1, t) ∂r −r 2 nvnH(rn, t) (3.5)
Furthermore, the inverse Hankel transform is (Tong and Wang, 2005; Bandelli and Rajagopal, 1995) v(r, t) = π 2 2 ∞ X n=1 r2 nJ02(R2rn)B(r, rn) J2 1(R1rn) − J02(R2rn) vnH(t) (3.6) and R2 Z R1 r log r R2 B(r, rn) dr = 2 πR1r3 n (3.7)
Multiplying Eq. (2.5) by rB(r, rn), integrating the result with respect to
r from R1 to R2, and using boundary condition (3.3)1 and identity (3.5), we
find that λ¨vnH(t) + (1 + αr2n) ˙vnH(t) + νr 2 nvnH(t) = 2f t πρrn t > 0 (3.8) From (3.2) it also results that
vnH(0) = ˙vnH(0) = 0 (3.9)
The solution to linear ordinary differential equation (3.8), with initial condi-tions (3.9), is given by vnH(t) = 2f πµr3 n h t − e q2nt−eq1nt q2n−q1n − 1 + αr2 n νr2 n 1 −q2ne q1nt−q1neq2nt q2n−q1n i (3.10) where q1n, q2n = −(1 + αr 2 n) ± p (1 + αr2 n)2−4νλrn2 2λ
Finally, applying the inverse Hankel transform formula and using Eq. (3.7), we find for the velocity field v(r, t) the simple expression
v(r, t) = f µ(t − λr)R1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] +πf µ ∞ X n=1 J2 0(R2rn)B(r, rn) rn[J2 1(R1rn) − J 2 0(R2rn)] (3.11) ·1 + αr 2 n νr2 n q2neq1nt−q1neq2nt q2n−q1n − eq2nt−eq1nt q2n−q1n or equivalently v(r, t) = f µ(t − λr)R1log r R2 − πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] 1 − λq 2 1neq 2nt−q2 2neq 1nt q2n−q1n (3.12)
3.2. Calculation of the shear stress
Solving Eq. (2.4)1 with respect to τ (y, t) and taking into account Eq.
(3.2)3, we find that τ (r, t) = µ λe −t λ t Z 0 eλs 1 + λr ∂ ∂s ∂rv(r, s) ds (3.13)
Substituting (3.12) into (3.13), we obtain, after lengthy but straightforward computations, a suitable form for the shear stress
τ (r, t) = fR1 r [t − λ(1 − e−λt)] +πf ν (1 − e −t λ) ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J02(R2rn)] +πf ν ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J 2 0(R2rn)] λ q2n−q1n hq2 2n(eq 1nt−e− t λ) 1 + λq1n (3.14) −q 2 1n(eq 2nt−e− t λ) 1 + λq2n + λrq1nq2n q2n(eq1nt−e− t λ) 1 + λq1n −q1n(e q2nt−e− t λ) 1 + λq2n i where e B(r, rn) = J1(rrn)Y1(R1rn) − J1(R1rn)Y1(rrn) (3.15)
Of course, Eq. (3.14) can further be processed to give the simple form
τ (r, t) = fR1 r [t − λ(1 − e−λt)] +πf ν ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J02(R2rn)] 1 − q2ne q1nt−q1neq2nt q2n−q1n (3.16) 4. Limiting cases 4.1. Maxwell fluid
Taking the limit of Eqs. (3.12) and (3.16) as λr → 0, we attain to the solutions vM(r, t) = f t µR1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] 1 − λq 2 5neq 6nt− q2 6neq 5nt q6n−q5n τM(r, t) = f R1 r [t − λ(1 − e−t λ)] +πf ν ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J02(R2rn)] 1 −q6ne q5nt−q5neq6nt q6n−q5n (4.1)
corresponding to a Maxwell fluid performing the same motion. Into the above relations q5n, q6n= −1 ± p 1 − 4νλr2 n 2λ
4.2. Second grade fluid
By now letting λ → 0 into Eqs. (3.12) and (3.16), the similar solutions
vSG(r, t) = f µ(t − λr)R1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] h 1 − (1 + αr2 n) exp − νr 2 nt 1 + αr2 n i τSG(r, t) = f tR1 r +πf ν ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J02(R2rn)] h 1 − exp− νr 2 nt 1 + αr2 n i (4.2)
corresponding to a second grade fluid are recovered. 4.3. Newtonian fluid
Finally, making λ and λr → 0 into Eqs. (3.12) and (3.16) or λ → 0 into
(4.1) respectively, λr→0 into (4.2), the solutions for a Newtonian fluid
vN(r, t) = f t µR1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] (1 − e−νr2nt) (4.3) τN(r, t) = f t R1 r +πf ν ∞ X n=1 J2 0(R2rn)B(r, re n) r2 n[J12(R1rn) − J02(R2rn)] (1 − e−νrn2t)
are obtained. Of course, for the last two cases (Newtonian and second grade fluids), the boundary condition obtained from (3.1) for λ → 0 is
τ (R1, t) = f t (4.4)
5. Connection with some similar results from the literature
The unsteady motion through an infinite annular region due to the inner cylinder that applies a constant longitudinal shear f to a second grade fluid has been studied by Bandelli and Rajagopal (1995). The velocity field that was obtained is v0SG(r, t) = f µR1log r R2 −πf µ ∞ X n=1 J2 0(R2rn)B(r, rn) rn[J2 1(R1rn) − J02(R2rn)] exp− νr 2 nt 1 + βr2 n (5.1)
where β = α1/µ, α1 is a material constant and rn are the same roots as before. Simple analysis shows that the two solutions v0
SG(r, t) and vSG(r, t), corresponding to the shear stress f , respectively f t on the boundary, are related by the integral relation
vSG(r, t) = t
Z 0
v0SG(r, s) ds (5.2)
if α = β = νλr. Of course, simple computations also show that
τSG(r, t) = t
Z 0
τSG0 (r, s) ds (5.3)
Consequently, the solutions vSG(r, t) and τSG(r, t), as well as vN(r, t) and
τN(r, t), corresponding to the motion of a second grade or Newtonian fluid due to the inner cylinder that applies a shear stress f t to the fluid, can immediately be obtained by simple integration if the similar solutions v0
SG(r, t), τSG0 (r, t),
v0
N(r, t) and τ
0
N(r, t) corresponding to a constant shear on the boundary, are known.
6. Numerical results and conclusions
The purpose of this note is to provide exact solutions for the velocity field and the shear stress corresponding to the axial flow of an Oldroyd-B fluid between two infinite circular cylinders, the inner cylinder being subject to a time-dependent shear stress and the outer one at rest. These solutions, obta-ined by means of the finite Hankel transforms, are presented under a series form in terms of the Bessel functions J0(·), J1(·), Y0(·) and Y1(·). Direct
com-putations show that they satisfy both the governing equations and all initial and boundary conditions. Furthermore, for λr → 0 or λ → 0, general
solu-tions (3.12) and (3.16) reduce to the corresponding solusolu-tions for Maxwell fluids and second grade fluids, respectively. The last solutions, as well as the general solutions, can easily be specialized to give the similar solutions for Newtonian fluids performing the same motion.
All solutions are presented as a sum between large-time and transient so-lutions. Simple analysis shows that
vLT(r, t) = f µ(t − λr)R1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] = vSGLT(r, t) (6.1) vM LT(r, t) = f t µR1log r R2 −πf µν ∞ X n=1 J2 0(R2rn)B(r, rn) r3 n[J12(R1rn) − J02(R2rn)] = vN LT(r, t)
Consequently, as expected, for large times the velocity of an Oldroyd-B flu-id can be well enough approximated by that of a second grade fluflu-id. The last relation shows that for Maxwell fluids the non-Newtonian effects on the fluid motion disappear in time. As regards the shear stresses, it is clearly that τLT(r, t) = τM LT(r, t) and τSGLT(r, t) = τN LT(r, t). This is not a surpri-se, because the shear stress on the inner cylinder is the same for Oldroyd-B and Maxwell fluids, respectively for the second grade and Newtonian fluids.
Finally, in order to reveal some relevant physical aspects of the obtained results, diagrams of the velocity field and the shear stress have been drawn against r for different values of t and material costants. Figures 1a and 1b clearly show that the velocity v(r, t), as well as the shear stress τ (r, t) in absolute value, is an increasing function of t. The influence of the relaxation and retardation times λ and λron the fluid motion is shown into Fig. 2 and 3. The velocity of the fluid, as well as the shear stress in absolute value, is a decreasing function of λ and λr. The effect of the kinematic viscosity, as it results from Figs. 4a and 4b, is opposite to that of λ and λr. The velocity of the fluid, as expected, increases for increasing ν.
Fig. 1. Profiles of the velocity v(r, t) and the shear stress τ (r, t) given by Eqs. (3.12) and (3.16), for R1= 0.5, R2= 0.9, f = −5, ν = 0.0357541, µ = 32, λ = 5, λr= 3
Fig. 2. Profiles of the velocity v(r, t) and the shear stress τ (r, t) given by Eqs. (3.12) and (3.16), for R1= 0.5, R2= 0.9, f = −5, ν = 0.0357541, µ = 32, λr= 3, t = 15 s,
and different values of λ
Fig. 3. Profiles of the velocity v(r, t) and the shear stress τ (r, t) given by Eqs. (3.12) and (3.16), for R1= 0.5, R2= 0.9, f = −5, ν = 0.0357541, µ = 32, λ = 7, t = 15 s,
and different values of λr
Fig. 4. Profiles of the velocity v(r, t) and the shear stress τ (r, t) given by Eqs. (3.12) and (3.16), for for R1= 0.5, R2= 0.9, f = −5, ρ = 895, λ = 5, λr= 3, t = 15 s, and
Acknowledgement
The author M. Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathe-matics, NED University of Engineering & Technology, Karachi-75270, Pakistan and also Higher Education Commission of Pakistan for generous support and facilitation of this research work.
The author Mehwish Rana remains highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan and also Higher Education Commission of Pakistan for generous supporting and facilitating this research work.
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Wybrane rozwiązania dokładne dla cieczy Oldroyda-B przy zadanej funkcji naprężeń stycznych zależnej od czasu
Streszczenie
Pole prędkości i pole rozkładu naprężeń stycznych wywołanych ruchem cieczy Oldroyda-B umieszczonej między dwoma koncentrycznymi cylindrami wyznaczono za pomocą transformaty Hankela. Przepływ cieczy wywołano zależnym od czasu naprę-żeniem stycznym od zewnętrznej ściany cylindra wewnętrznego. Uzyskane rozwiązanie dokładne, ujęte w formie rozwinięcia w szereg, może łatwo być zastosowane dla przy-padków szczególnych cieczy Maxwella, cieczy drugiego stopnia i nienewtonowskich przy tych samych warunkach przepływu. Na zakończenie rozważań, przedstawiono graficznie charakterystyki ruchu cieczy i wpływ parametrów materiałowych na jej zachowanie.
